A New Division Algebra Representation of $E_6$

Abstract

We construct the well-known decomposition of the Lie algebra ${\mathfrak{e}}_8$ into representations of ${\mathfrak{e}}_6\oplus \mathfrak{su}(3)$ using explicit matrix representations over pairs of division algebras. The minimal representation of ${\mathfrak{e}}_6$, namely the Albert algebra, is thus realized explicitly within ${\mathfrak{e}}_8$, with the action given by the matrix commutator in ${\mathfrak{e}}_8$, and with a natural parameterization using division algebras. Each resulting copy of the Albert algebra consists of anti-Hermitian matrices in ${\mathfrak{e}}_8$, labeled by imaginary (split) octonions. Our formalism naturally extends from the Lie algebra to the Lie group $E_6\subset E_8$.

9 pages